Differential transcendence criteria for second-order linear difference equations and elliptic hypergeometric functions

Arreche, Carlos E.; Dreyfus, Thomas; Roques, Julien

Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 147-169.

Dans cet article, nous développons des critères généraux garantissant la transcendance différentielle d’une solution non nulle donnée d’une équation aux différences du deuxième ordre. Ces critères s’appliquent à de nombreuses équations, telles que les équations aux différences finies, les équations aux $q$-différences, les équations de Mahler, ou encore les équations aux différences elliptiques. Notre approche repose sur la théorie de Galois des équations aux différences. En guise d’application, nous démontrons que la plupart des fonctions hypergéométriques elliptiques sont différentiellement transcendantes.

We develop general criteria that ensure that any non-zero solution of a given second-order difference equation is differentially transcendental, which apply uniformly in particular cases of interest, such as shift difference equations, $q$-dilation difference equations, Mahler difference equations, and elliptic difference equations. These criteria are obtained as an application of differential Galois theory for difference equations. We apply our criteria to prove a new result to the effect that most elliptic hypergeometric functions are differentially transcendental.

Classification : 39A06, 12H05
Keywords: Linear difference equations, difference Galois theory, elliptic curves, differential algebra
Mot clés : Équations aux différences linéaires, théorie de Galois aux différences, courbe elliptiques, algèbre différentielle

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